Incoming mails are displayed in both INBOX and „Name-of-Mailbox“. It is not any sort of „intelligent mailbox“ or similar. There is no rule pointing to that mailbox.
You can not rename or delete that mailbox, even if it`s empty. Neither in any mail-app nor in the webmailer-interface.

There is no sub-folder for „Name-of-Mailbox“ (.mdir) on the server / within the User.macnt.